Classical models for the growth and spread of introduced species
track the front of an expanding wave of population density. Models
are typically parabolic partial differential equations and related integral
formulations. One method to infer the speed of the expanding wave is
to equate the speed of spread of the nonlinear system with the speed of
spread of a related linear system. When these two speeds coincide we say that the
spread rate is linearly predictable. While many spread rates are linearly
predictable, some notable cases are not, such as those involving competition between
multiple species.

Hans Weinberger's work has impacted the theory of linear predictability,
both for single-species and for multi-species models. I will review
some of this theory, from the perspective of a mathematical ecologist
interested in applying the theory to biology. In my talk I will apply some of the results
to real biological problems, including species competition, spread of disease and
population dynamics of stream ecosystems.

In this talk we will survey several papers (listed
below) by
Hans Weinberger dealing with numerical and approximation
issues. We have
divided them into three categories: (i) approximation of
eigenvalues; (ii)
approximation theory issues; and (iii) error bounds for
iterative methods for
matrix inversion.

The seven papers listed are only a small part of Hans’ work—but
they
were very influential. We, of course, cannot discuss any of
these papers
in detail, but will instead concentrate on those results that
are especially
insightful and elegant.